1. The Erik Jonsson School of Engineering and Computer Science Chapter 2 pp. 49-100 William J. Pervin The University of Texas at Dallas Richardson, Texas 75083

2. The Erik Jonsson School of Engineering and Computer Science Chapter 2 Discrete Random Variables

3. The Erik Jonsson School of Engineering and Computer Science Chapter 2 2.1 Definitions: A random variable (X) consists of a experiment with a probability measure P[.] defined on a sample space S and a function X that assigns a real number X(s) to each outcome s  S.

4. The Erik Jonsson School of Engineering and Computer Science Chapter 2 Shorthand notation: {X=x} ≡ {s  S | X(s) = x} Discrete vs. Continuous RVs

5. The Erik Jonsson School of Engineering and Computer Science Chapter 2 2.2 Probability Mass Function: The PMF (P X ) of the discrete random variable X is P X (x) = P[X=x] = P[{s  S | X(s) = x}]

6. The Erik Jonsson School of Engineering and Computer Science Chapter 2 Theorem: For any discrete random variable X with PMF P X and range S X : 1. (  x) P X (x) ≥ 0 2. Σ x  S X P X (x) = 1 3. (  B  S X ) P[X  B] = P[B] = Σ x  B P X (x)

7. The Erik Jonsson School of Engineering and Computer Science Chapter 2 2.3 Families of Discrete RVs Bernoulli (p) RV: (0 < p < 1) P X (x) = 1-p if x=0, p if x=1, 0 otherwise (Two outcomes)

8. The Erik Jonsson School of Engineering and Computer Science Chapter 2 Geometric (p) RV: (0 < p < 1) P X (x) = (1-p) x-1 p, x=1,2,…; 0 otherwise (Number to first success) Binomial (n,p) RV: (0 < p < 1; n = 1,2,…) P X (x) = C(n,x)p x (1-p) n-x (Number of successes in n trials) (Note: Binomial(1,p) is Bernoulli)

9. The Erik Jonsson School of Engineering and Computer Science Chapter 2 Pascal (n,p) RV: (0 < p < 1; n = 1,2,…) P X (x) = C(x-1,n-1)p k (1-p) x-n (Number to n successes) (Note: Pascal(1,p) is Geometric) Discrete Uniform (m,n) RV: (m<n integers) P X (x) = 1/(n-m+1) for x=m,m+1,…,n; 0 otherwise

10. The Erik Jonsson School of Engineering and Computer Science Chapter 2 Poisson (α) RV: (α > 0) P X (x) = α x e -α /x! for x=0,1,…; 0 otherwise (Arrivals: α = λT)

11. The Erik Jonsson School of Engineering and Computer Science Chapter 2 2.4 Cumulative Distribution Function The CDF (F X ) of a random variable X is F X (x) = P[X ≤ x]

12. The Erik Jonsson School of Engineering and Computer Science Chapter 2 For any discrete random variable X with range S X = {x 1 ≤ x 2 ≤ …}: the CDF (F X ) is monotone non-decreasing from 0 to 1, with jump discontinuities of height P X (x i ) at each x i  S X and constant between the jumps.

13. The Erik Jonsson School of Engineering and Computer Science Chapter 2 F X (b) – F X (a) = P[a < X ≤ b]

14. The Erik Jonsson School of Engineering and Computer Science Chapter 2 2.5 Averages Statistics: mean, median, mode, … Parameter of a model: mode, median Expected Value of X = E[X] = μ X = Σ x  S X xP X (x)

15. The Erik Jonsson School of Engineering and Computer Science Chapter 2 E[X] = p if X is Bernoulli (p) RV E[X] = 1/p if X is geometric (p) RV E[X] = α if X is Poisson (α) RV E[X] = np if X is binomial (n,p) RV E[X] = k/p if X is Pascal (k,p) RV E[X] = (m+n)/2 if X is discrete uniform (m,n) RV

16. The Erik Jonsson School of Engineering and Computer Science Chapter 2 Note: Poisson PMF is limiting case of binomial PMF.

17. The Erik Jonsson School of Engineering and Computer Science Chapter 2 2.6 Functions of a Random Variable Derived RV Y = g(X) for RVs when y = g(x) for values P Y (y) = Σ x:g(x)=y P X (x)

18. The Erik Jonsson School of Engineering and Computer Science Chapter 2 2.7 Expected Value of a Derived RV If Y = g(X) then E[Y] = μY = Σ x  S X g(x)P X (x) For any RV X: E[X-μ X ] = 0 and E[aX + b] = aE[X] + b

19. The Erik Jonsson School of Engineering and Computer Science Chapter 2 E[X 2 ] = Σ x  S X x 2 P(x)

20. The Erik Jonsson School of Engineering and Computer Science Chapter 2 2.8 Variance and Standard Deviation Var[X] = E[(X-μ X ) 2 ] σ X = sqrt(Var[X]) Var[X] = E[X 2 ] – (E[X]) 2 = E[X 2 ] – μ X 2

21. The Erik Jonsson School of Engineering and Computer Science Chapter 2 Moments of a RV X: n th moment: E[X n ] n th central moment: E[(x – μ X ) n ] Theorem: Var[aX + b] = a 2 Var[X]

22. The Erik Jonsson School of Engineering and Computer Science Chapter 2 Var[X] = p(1-p) if X is Bernoulli (p) RV Var[X] = (1-p)/p 2 if X is geometric (p) RV Var[X] = α if X is Poisson (α) RV Var[X] = np(1-p) if X is binomial (n,p) RV Var[X] = k(1-p)/p 2 if X is Pascal (k,p) RV Var[X] = (n-m)(n-m+2)/12 if X is discrete uniform (m,n) RV

23. The Erik Jonsson School of Engineering and Computer Science Chapter 2 2.9 Conditional PMF P X|B (x) = P[X=x|B] 2.10 MATLAB